Problem: How many distinct prime factors does the sum of the positive divisors of $400$ have?
Explanation: First, we find what the sum of the divisors of $400$ is.

The prime factorization of $400$ is $2^4 \cdot 5^2$. Therefore, the sum of the divisors is $(1+2+2^2+2^3+2^4)(1+5+5^2) = 31 \cdot 31$. To see why the expression on the left-hand side gives the sum of the divisors of 400, note that if you distribute (without simplifying), you get 15 terms, each divisor of $2^4\cdot 5^2$ appearing exactly once.

Since $31$ is a prime number, the sum of the positive divisors of $400$ only has $\boxed{1}$ prime factor.